Rounds in a combinatorial search problem

TL;DR

This paper investigates the minimum number of questions needed in a combinatorial search problem to find excellent elements in a set, confirming a conjecture and extending results to multiple rounds and multiple elements.

Contribution

We prove a conjecture of Katona by establishing the exact number of queries needed in the r-round version of the problem, and extend bounds to finding multiple excellent elements.

Findings

In the r-round version, the number of queries needed is rn^{1/r}+O(1).

The result confirms Katona's conjecture and is sharp for fixed r.

Bounds are provided for finding at least d excellent elements.

Abstract

We consider the following combinatorial search problem: we are given some excellent elements of $[n]$ and we should find at least one, asking questions of the following type: "Is there an excellent element in $A \subset [n]$?". G.O.H. Katona proved sharp results for the number of questions needed to ask in the adaptive, non-adaptive and two-round versions of this problem. We verify a conjecture of Katona by proving that in the $r$-round version we need to ask $rn^{1/r}+O(1)$ queries for fixed $r$ and this is sharp. We also prove bounds for the queries needed to ask if we want to find at least $d$ excellent elements.